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MA-302 Advanced Calculus 8 Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore Aug-Dec 2002 MA-302 Advanced Calculus 8. Inverse function theorem Carl Gustav Jacob Jacobi† (1804-1851) The exercises 8.1 and 8.2 are only to get practice, their solutions need not be submitted. 8.1. Determine at which points a of the domain of definition G the following maps F have differentiable inverse and give the biggest possible open subset on which F define a diffeomorphism. a). F : R2 → R2 with F(x,y) := (x2 − y2, 2xy) . (In the complex case this is the function z → z2.) b). F : R2 → R2 with F(x,y) := (sin x cosh y,cos x sinh y). (In the complex case this is the function z → sin z.) c). F : R2 → R2 with F(x,y) := (x+ y,x2 + y2) . d). F : R3 → R3 with F(r,ϕ,h) := (r cos ϕ,rsin ϕ,h). (cylindrical coordinates) × 2 2 3 3 e). F : (R+) → R with F(x,y) := (x /y , y /x) . xy f). F : R2 → R2 with F(x,y) := (x2 + y2 ,e ) . 8.2. Show that the following maps F are locally invertible at the given point a, and find the Taylor-expansion F(a) of the inverse map at the point upto order 2. a). F : R3 → R3 with F(x,y,z) := − (x+y+z), xy+xz+yz,−xyz at a =(0, 1, 2) resp. a =(−1, 2, 1). ( Hint : The problem consists in approximating the three zeros of the monic polynomial of degree 3 whose coefficients are close to those of X(X−1)(X−2) resp. (X+1)(X−2)(X−1). – In the general case the map n n n n n : K →K , which maps the point (x1,...,xn)∈ K to the coefficient tuple (−s1,s2,...,(−1) sn) of the n n−1 n monic polynomial (X−x1) ···(X−xn) = X − s1X +···+(−1) sn which has the zeros x1,...,xn, n is locally invertible precisely at the points a = (a ,...,an) ∈ K which has distinct components. Upto a 1 n+1 n (− )( 2 ) (x −x ) = (− ) (x ,...,x ) sign the Jacobian determinant 1 1≤i<j≤n i j 1 V 1 n of n is a Vandermonde’s determinant. To find the zeros of polynomials of degree n, one need to invert the map n.) xy b). F : R3 → R3 with F(x,y,z) := (x2 + y3z,e , sin x + sin y) at a = (0,π/2, 0) . x+y y c). F : R2 → R2 with F(x,y) = (e ,xe ) at a = (0, 0) . 8.3. Let G ⊆ V be an open subset and let F : G → V be a continuously differentiable map with a fixed point a ∈G. If 1 is not an eigenvalue of the total differential (DF)a : V →V of F at the point a, then show that a is an isolated fixed point of F (i.e. there exists an open neighbourhood of a,in which F has no other fixed point). 8.4. Find a (real) Cω-diffeomorphism F : G→G for the following open subsets G⊆V , G ⊆W. n a). G:= R , G := B(x0 ; r), r>0 and W is an inner product space with DimRW = n. G = Rn G = (a ,b ) ×···× (a ,b ) ⊆ Rn −∞≤a <b ≤∞ i = ,...,n b). : , : 1 1 n n , i i , 1 . G = (x ,...,x )∈ Rn x2 +···+x2 < G = (− , )n ⊆ Rn c). : 1 n 1 n 1 , : 1 1 . D. P. Patil January 24, 2005 ,12:33 p.m. 2 Advanced Calculus ; Jan-April 2002 ; 8. Inverse function theorem n d). G := R −{0} , G := B(x0 ; r) − B(y0 ; s),0≤ s ≤ s +y0 −x0 <r and W is an inner product space with DimRW = n. e). G:= R2, G := (x, y) ∈ R2 |xy|<1 . n f). G:= R , G is the interior of a non-degenerate n-simplex in W, where n:= DimRW. (Induction on n.) 8.5. (Cycloide-coordinates) Themap (a, σ ) → a(σ − sin σ,1 − cos σ) isaCω- × × × diffeomorphism of R+ × (0, 2π) onto R+ × R+. (The coordinate-lines a = a0 = const. are cycloids.) 8.6. (Torus-coordinates) LetR>0. The map T : (r,ϕ,ψ)→ (R−r cos ψ)cos ϕ,(R−r cos ψ)sin ϕ,rsin ψ isaCω-diffeomorphism of (0,R) × (0, 2π)2 onto the slit open torus. The coordinate-system (r,ϕ,ψ)is orthogonal. Compute the Jacobian determinant of T and give the Laplace-Operator in the coordinates r, ϕ, ψ. 8.7. (Cone-coordinates) Let V be an Euclidean vector space. The map (v, α) → (v, v tan α) isaCω-diffeomorphism of V \{0} × (−π/2 ,π/2) onto V \{0} ×R. Compute the Jacobian determinant of this map. (The coordinate hyperplanes α =α0 = const. are cones (without vertex) . ) D. P. Patil January 24, 2005 ,12:33 p.m. Advanced Calculus ; Jan-April 2002 ; 8. Inverse function theorem 3 8.8. (Power-coordinates)LetI be a finiet index set. For a matrix α =(αij ) ∈ MI (R), let Fα ω × I × I αij denote the C - map (R+) →(R+) with Fα (tj )j∈I = j∈I tj . i∈I −1+ i∈I αij a). The Jacobian determinant of Fα is J(Fα ; t) = (Det α) t . Show that Fα is a j∈I j diffeomorphism if and only if α is invertible. b). The map α → Fα is an (injektive) homomorphism of the group GLI (R) in the group of ω × I C -diffeomorphisms of (R+) into itself. × I ( Hint : One can clearly see the map Fα, in the coordinates xi := ln ti , i ∈I,on(R+) ). † Carl Gustav Jacob Jacobi (1804-1851) was born on 10 Dec 1804 in Potsdam, Prussia (now Germany) and died on 18 Feb 1851 in Berlin, Germany. Carl Jacobi came from a Jewish family but he was given the French style name Jacques Simon at birth. His father, Simon Jacobi, was a banker and his family were prosperous. Carl was the second son of the family, the eldest being Moritz Jacobi who eventually became a famous physicist. There was a sister, Therese Jacobi, and a third brother, Eduard Jacobi, who was younger than Carl. Eduard did not pursue an academic career, but followed instead his father’s profession as a banker. Jacobi’s early education was given by an uncle on his mother’s side, and then, just before his twelfth birthday, Jacobi entered the Gymnasium in Potsdam. While still in his first year of schooling, he was put into the final year class and hence when he was still only 12 years old yet he had reached the necessary standard to enter university. The University of Berlin, however, did not accept students below the age of 16, so Jacobi had to remain in the same class at the Gymnasium in Potsdam until the spring of 1821. Of course, Jacobi pressed on with his academic studies despite remaining in the same class at school. He received the highest awards for Latin, Greek and history but it was the study of mathematics which he took furthest. By the time Jacobi left school he had been undertaking research on his own attempting to solve quintic equations by radicals. Jacobi entered the University of Berlin in 1821, he chose mathematics, but this did not mean that he could attend high level courses in mathematics for at this time the standard of university education in mathematics in Germany was rather poor. As he had done at the Gymnasium, Jacobi had to study on his own reading the works of Lagrange and other leading mathematicians. By the end of academic year 1823-24 Jacobi had passed the examinations necessary for him to be able to teach mathematics, Greek, and Latin in secondary schools. In 1825, he was offered a teaching post at the Joachimsthalsche Gymnasium, one of the leading schools in Berlin. He had submitted his doctoral dissertation to the University of Berlin even before he received the offer of the teaching post, and he was allowed to move quickly to work on his habilitation thesis. Jacobi presented a paper concerning iterated functions to the Academy of Sciences in Berlin in 1825. However, the referees did not consider the results worth publishing and indeed the paper was not published by the Berlin Academy of Sciences. The paper was published eventually in 1961. Although this was not the best start for the young Jacobi, it did not hold him back for long and his publication record over the following years would be quite remarkable for both the number and quality of the works. Around 1825 Jacobi changed from the Jewish faith to become a Christian which now made university teaching possible for him. By the academic year 1825-26 he was teaching at the University of Berlin. However prospects in Berlin were not good so, after taking advice from colleagues, Jacobi moved to the University of Königsberg. There he joined Franz Neumann, who had also received his doctorate from Berlin in 1825, and Bessel who was the professor of astronomy at Königsberg. Jacobi had already made major discoveries in number theory before arriving in Königsberg. He now wrote to Gauss to tell him of the results on cubic residues which he had obtained, having been inspired by Gauss’s results on quadratic and biquadratic residues. Gauss was impressed, so much so that he wrote to Bessel to obtain more information about the young Jacobi. But Jacobi also had remarkable new ideas about elliptic functions (as Abel did quite independently and at much the same time).
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